A study of discrete and integral transforms with logarithmic separable kernels

In this thesis, we will be examining different classes of discrete and integral transforms. We start with a general class of integral transforms which include those logarithmic separable kernels. Transforms with logarithmic separable kernels include the Fourier transform, the Laplace transform and the Mellin transform. The shifting and convolution properties for this class of transforms are examined, and sufficient conditions which guarantee the existence of the convolution formula will be given. It will be shown that a subclass of these integral operators are injective and an inversion formula will be presented on some class of continuously differentiable functions. We will apply these results to second-order differential equations to obtain new analytical solutions to these equations and compare these to a numerical solution.